More than 35 years — half a lifetime! — have passed since “The Exorcist,” and horror movies about casting the devil out of people have become quaint. So it’s nice to see a little jolt put back in the old formula with “The Last Exorcism,” a tense and generally understated chiller that might creep you right the hell out.

It’s presented as a fake documentary (speaking of quaint movie premises) about a charismatic evangelical preacher named Cotton Marcus (Patrick Fabian), a second-generation preacher in Baton Rouge who’s as much of a showman as he is a minister. He’s made a comfortable living for himself and his family, supplemented by performing exorcisms for those in need. He doesn’t really believe in demonic possession, but his smoke-and-mirrors routine makes people feel like they’ve been exorcised, so what’s the harm?

Rev. Marcus has decided to call it quits, though, and has permitted the filmmaker, Iris Reisen (Iris Bahr), to document — wait for it — his last exorcism. Accompanied by a cameraman, they set out for an isolated farm in rural Louisiana, where a widower named Louis Sweetzer (Louis Herthum) lives with his teenage children, Nell (Ashley Bell) and Caleb (Caleb Landry Jones). One of the Sweetzers might currently be occupied by Beelzebub!

And I shall tell you no more of the plot. What I shall tell you instead is that screenwriters Huck Botko and Andrew Gurland (the duo behind another clever mockumentary, “Mail Order Wife”) have constructed a story full of suspense and mystery. It’s easy to predict that Rev. Marcus, who doesn’t believe in demonic possession, will come in contact with something that makes him reevaluate this position, but “The Last Exorcism” keeps us guessing on the particulars. Who or what is behind the unusual events at the Sweetzer farm? Is the explanation supernatural, human, or a combination of the two? The movie maintains its secrets right up until the end, without having to cheat or deceive us, and we’re genuinely interested in what becomes of these characters, too.

The director, Daniel Stamm, is another fake-doc veteran; his “A Necessary Death” purported to show a trio of young filmmakers searching for a suicidal person whose death they could document. “The Last Exorcism” isn’t quite that morbid, but it is marvelously creepy. Except for some musical scoring that may have been ill-advised, Stamm maintains a startlingly realistic approach, with nothing to ruin the illusion that this footage was shot by a real documentary crew. (Well, except that you might recognize a couple of the actors.)

You could claim that the story goes a bit off the rails at the end and I probably wouldn’t argue with you. But there are only so many ways this scenario can play out, and I admire the gutsiness of choosing this one, however outlandish it may be. For the most part, the scares are subtle: a look at the camera here, an unexpected appearance there, a strange sound from behind that door. Whatever is going on at the Sweetzer farm, I hope it stays there.